Note that due to the first condition H2, by virtue of Theorem 3.4 [1]} the operator \(W\)  is closable.
Let us show that \(\tilde{W}\)  is sectorial. By virtue of condition H2, we get
\(\mathrm {Re}(\tilde{W}f,f)_{\mathfrak {H}}\ge C_{2}\Vert f\Vert ^{2}_{\mathfrak {H_{+}}}\ge C_{2}\varepsilon \Vert f\Vert ^{2}_{\mathfrak {H}_{+}}+\frac{C_{2}(1-\varepsilon )}{C_{0}}\Vert f\Vert ^{2}_{\mathfrak {H}};\) 
\(\mathrm {Re}(\tilde{W}f,f)_{\mathfrak {H}} -k|\mathrm {Im}(\tilde{W}f,f)_{\mathfrak {H}}|\ge (C_{2}\varepsilon -k C_{1})\Vert f\Vert ^{2}_{\mathfrak {H}_{+}} +\frac{C_{2}(1-\varepsilon )}{C_{0}}\Vert f\Vert ^{2}_{\mathfrak {H}}=\frac{C_{2}(1-\varepsilon )}{C_{0}}\Vert f\Vert ^{2}_{\mathfrak {H}},\) 
